Title Axial symmetry and transverse trace-free tensors in numerical relativity Author(s) Conboye, Rory P. A. Publication date 2012-02 Original citation Conboye, Rory P.A. 2012. Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity. PhD Thesis, University College Cork. Type of publication Doctoral thesis Rights © 2012, Rory Patrick Albert Conboye http://creativecommons.org/licenses/by-nc-nd/3.0/ Item downloaded http://hdl.handle.net/10468/527 from Downloaded on 2017-02-12T08:07:03Z Axial Symmetry and Transverse Trace-Free Tensors in Numerical Relativity Rory Patrick Albert Conboye NATIONAL UNIVERSITY OF IRELAND, CORK (UNIVERSITYCOLLEGECORK) Faculty of Science Department of Physics Thesis submitted for the degree of Doctor of Philosophy February 2012 Supervisor: Prof. Niall O’Murchadha Head of Department: Prof. John McInerney Contents Contents Abstract . .v Acknowledgements . vi Introduction . vii 1 General Relativity and Differential Geometry 1 1.1 Relativity Theory . .2 1.1.1 From Special to General Relativity . .2 1.1.2 Notation Conventions . .3 1.2 Background Mathematics . .4 1.2.1 Manifolds, Vectors and Tensors . .4 1.2.2 The Metric Tensor . .9 1.2.3 Derivatives on a Manifold . 11 1.2.4 Coordinate Changes and Curvature . 17 1.2.5 Killing Vectors and the Frobenius Theorem . 21 1.3 General Relativity . 25 1.3.1 The Special Theory of Relativity . 25 1.3.2 The General Theory of Relativity . 27 1.3.3 Schwarzschild Solution . 32 1.3.4 Kerr Solution . 36 2 3 + 1 Numerical Relativity 39 2.1 The 3 + 1 Formalism . 40 2.1.1 Space-time Foliations . 41 2.1.2 Coordinate Adaptations . 42 2.1.3 Extrinsic Curvature . 44 2.1.4 Einstein Constraint Equations . 45 2.1.5 Evolution Equations . 47 2.1.6 The Cauchy Problem . 49 2.1.7 Exact Solutions in the 3 + 1 Formalism . 50 2.2 Initial Conditions . 51 2.2.1 Conformal Transformations of the Spatial Metric . 51 2.2.2 Transverse and Trace-Free Decomposition of Extrinsic Curvature 53 2.2.3 Conformal Transverse Trace-free Decomposition . 55 2.2.4 Conformal Thin Sandwich Decomposition . 58 2.2.5 Bowen-York Extrinsic Curvature . 60 2.3 Gauge Conditions . 62 2.3.1 Geodesic Slicing . 62 2.3.2 Maximal and Constant Mean Curvature Slicing . 63 2.3.3 Harmonic Coordinates and “1+Log” Slicing . 65 2.3.4 Minimal Distortion Shift Conditions . 66 2.4 BSSN Reformulation . 69 Axial Symmetry and Transverse Trace-Free i R. Conboye Tensors in Numerical Relativity Contents 2.4.1 Conformal Transformation . 69 2.4.2 Conformal Connection Functions . 70 2.4.3 BSSN Equations . 71 3 Axially-Symmetric Transverse Trace-Free Tensors 72 3.1 Previous Work . 74 3.1.1 Baker & Puzio ........................... 74 3.1.2 Brandt & Seidel .......................... 75 3.1.3 Sergio Dain ............................. 75 3.1.4 Transverse and Trace-Free Nature of Sergio Dain ........ 77 3.1.5 Testing of Solution Sa ....................... 79 3.1.6 Comparing Sergio Dain with Baker & Puzio .......... 81 3.2 Flat Space Scalar Potentials . 82 3.2.1 Spherical Coordinates . 82 3.2.2 Cylindrical Coordinates . 87 3.2.3 Relating the Coordinate Potentials . 90 3.3 General Space Scalar Potentials . 96 3.3.1 Transverse Trace-Free Tensor with Brill Metric . 96 3.3.2 Time-Rotation Potential . 98 3.3.3 Equations for Tensors Free of Time-Rotation Symmetry . 99 3.3.4 Setting q(ρ, z) to One . 103 3.4 Relating Flat Space Potentials to Bowen-York Curvature . 104 3.4.1 Bowen-York Curvature with Angular Momentum Only . 104 3.4.2 Bowen-York Curvature with Linear Momentum Only . 106 3.4.3 Combined Angular and Linear Momentum . 108 3.5 Regularity of Tensors at the Origin . 109 3.5.1 Choice of Scalar Potential Y ................... 109 3.5.2 Choice of Scalar Potential X ................... 110 3.5.3 Tensor with Specified Potentials X and Y ............ 112 3.6 Spherically-Symmetric Tensor Products . 113 3.6.1 Time-Rotation Symmetry . 114 4 Kerr and Bowen-York Extrinsic Curvatures 115 4.1 Kerr and Bowen-York Background . 116 4.1.1 Bowen & York ........................... 116 4.1.2 Brandt & Seidel .......................... 116 4.1.3 Gleiser, Nicasio, Price & Pullin ................. 117 4.1.4 Garat & Price ........................... 117 4.1.5 Dain, Lousto & Takahashi ..................... 118 4.2 Extrinsic Curvatures and Derivatives . 119 4.2.1 Kerr Extrinsic Curvature . 119 4.2.2 Derivatives of the Curvature Components of Kerr . 120 4.2.3 Bowen-York Curvature . 121 Axial Symmetry and Transverse Trace-Free ii R. Conboye Tensors in Numerical Relativity Contents 4.3 Comparing the Curvatures of Kerr and Bowen-York . 123 4.3.1 Tensor Difference . 123 4.3.2 Flat Divergence of Tensor Difference . 125 5 Brill Wave Mass-Energy Approximations 128 5.1 Brill Wave Mass Evaluation . 129 5.1.1 Dieter Brill ............................. 129 5.1.2 Solving the Conformal Factor . 132 5.2 Numerical Approximation of Brill Wave Mass . 133 5.2.1 Deriving a Mass Expression for a Function q(ρ, z) ........ 133 5.2.2 Simplification of Lichnerowicz Equation . 135 5.2.3 Numerical Approximations . 137 6 Concluding Remarks 139 6.1 Axially-Symmetric Transverse Trace-Free Tensors . 140 6.2 Kerr and Bowen-York Extrinsic Curvatures . 142 6.3 Brill Wave Mass-Energy . 143 A Kerr Extrinsic Curvature Calculations 144 A.1 Extrinsic Curvature for the Natural Kerr Slicing . 145 A.1.1 Kerr Space-Time Metric in Boyer-Lindquist Coordinates . 145 A.1.2 The Inverse of the Kerr Metric . 146 A.1.3 Coordinate Derivatives of the Metric Components . 150 A.1.4 Lapse, Shift and Spatial Metric . 152 A.1.5 Evaluation of the Extrinsic Curvature . 153 A.1.6 Calculation of Individual Components . 157 A.1.7 Extrinsic Curvature with Raised Indices . 160 A.2 Derivatives of Extrinsic Curvature Components . 161 A.2.1 Derivative of Kφr, with respect to r ............... 162 A.2.2 Derivative of Kφθ, with respect to θ ............... 166 A.3 Einstein Constraints . 169 A.3.1 Hamiltonian Constraint . 169 A.3.2 Scalar Form of Extrinsic Curvature . 170 A.3.3 Momentum Constraint . 171 A.3.4 Evaluation of φ Divergence Term . 177 B Mathematica Calculations for Brill Mass-Factor 181 B.1 Solving Ordinary Differential Equation . 182 B.2 Graphing Mass-Factor . 183 List of Symbols 184 References 191 Axial Symmetry and Transverse Trace-Free iii R. Conboye Tensors in Numerical Relativity I, Rory Patrick Albert Conboye, certify that this thesis is my own work and I have not obtained a degree in this university or elsewhere on the basis of the work submitted in this thesis. Rory Patrick Albert Conboye Axial Symmetry and Transverse Trace-Free iv R. Conboye Tensors in Numerical Relativity Abstract Abstract Transverse trace-free (TT ) tensors play an important role in the initial conditions of numerical relativity, containing two of the component freedoms. Expressing a TT tensor entirely, by the choice of two scalar potentials, is not a trivial task however. Assuming the added condition of axial symmetry, expressions are given in both spherical and cylindrical coordinates, for TT tensors in flat space. A coordinate rela- tion is then calculated between the scalar potentials of each coordinate system. This is extended to a non-flat space, though only one potential is found. The remaining equations are reduced to form a second order partial differential equation in two of the tensor components. With the axially symmetric flat space tensors, the choice of potentials giving Bowen- York conformal curvatures, are derived. A restriction is found for the potentials which ensure an axially symmetric TT tensor, which is regular at the origin, and conditions on the potentials, which give an axially symmetric TT tensor with a spherically symmetric scalar product, are also derived. A comparison is made of the extrinsic curvatures of the exact Kerr solution and numerical Bowen-York solution for axially symmetric black hole space-times. The Brill wave, believed to act as the difference between the Kerr and Bowen-York space-times, is also studied, with an approximate numerical solution found for a mass-factor, under different amplitudes of the metric. Axial Symmetry and Transverse Trace-Free v R. Conboye Tensors in Numerical Relativity Acknowledgements Acknowledgements I would firstly like to acknowledge the guidance and support given by my super- visor Prof. Niall Ó Murchadha throughout this work. His patience and contagious enthusiasm have been invaluable. I would like to thank in particular Mark Hannam, Patrick Tuite, Padraig MacCar- ron, Rob Sheehan and Dan Hurley for many helpful discussions, both on topic and off, and to all of my family and friends, for your understanding and support. A special thanks must go to my parents Anna and Tony Conboye, and to Caroline Browne, for their consistent support and care, and for careful reading of many different drafts of this thesis. I must thank my examiners Dr. Denise Gabuzda, and Prof. Edward Malec, for their time and effort, and for helpful hints and corrections that have certainly improved the final version of this thesis. I would also like to acknowledge the support of the Science Foundation of Ireland for the funding which covered this project. Axial Symmetry and Transverse Trace-Free vi R. Conboye Tensors in Numerical Relativity Introduction Introduction This thesis is broken into five main chapters. The first two provide background material, with the remaining three covering the main work of the thesis. These are followed by a conclusions chapter, appendices providing specific calculations for chapter 4 and Mathematica input for chapter 5, and a list of the symbols used in the thesis, with descriptions and, where necessary, references to pages and equations.